As investigated in previous works, \(\phi \)  follows a power-law relationship, \((\phi _{c}-\phi ) \propto (t_{f}/\mathcal {T})^{\alpha }\) , where \(\alpha \)  is a scaling parameter. In dry granular systems, \(t_{f}/\mathcal {T}=t_{dry}/\mathcal {T}=2I/\sqrt{3}\)  and \(\alpha =1.0\)  [1]}, [2]}. As shown in Fig.REF (a), as decreasing the viscosity of the fluid, until the fluid effect is nearly nil, \((\phi _{c}-\phi ) \propto I\)  as shown in Fig.REF (a). In a submerged condition, when the fluid effect is dominant, \(t_{f}/\mathcal {T}=t_{sub}/\mathcal {T}=12I_{v}\)  and \(\alpha =0.5\) , as shown in Fig.REF (b), with the increase of the viscosity of fluid above a threshold (here \(\eta _{f}=1\)  g/(cm\(\cdot \) s)), data collapse to \((\phi _{c}-\phi ) \propto (I_{v})^{0.5}\) , which is also obtained in Refs.[3]}, [4]}. However, when the fluid viscosity varies in a wide range, the fluid effect and contact effect are both significant, \(\alpha \)  varies with the viscosity of fluid [5]}. Although, \(G=t_{f}/\mathcal {T}\)  naturally contains the varying effects of fluid and grain contact in different conditions, and smoothly transforms from \(I=t_{dry}/\mathcal {T}\)  to \(I_{v}=t_{sub}/\mathcal {T}\) , the exponents of \(\alpha \)  for \((\phi _{c}-\phi ) \propto (t_{f}/\mathcal {T})^{\alpha }\)  based rheological law from dry to very viscous regime is still hold an unclear decay from 1 to 0.5 that leads to \((\phi _{c}-\phi ) \propto G\)  being not universal as shown in Fig.REF (b). On the contrary, as we have predicted in the previous section, \(\mathcal {G}\)  obtains a good collapse of all data, which shows that the dimensionless length scale \(\mathcal {G}\)  can well describe the change of both inertial and hydraulic effects during flow regime transitions.
Thus, we have
\(\phi (\mathcal {G})/\phi _{c}=\frac{1}{1+b\sqrt{\mathcal {G}}},\) 
